Science —

D-Wave’s quantum optimizer might be quantum after all

Quantum optimizer's behavior clearly isn't classical.

Quantum optimizer manufacturer D-Wave Systems has been gaining a lot of traction recently. They've sold systems to Lockheed Martin and Google, and started producing results showing that their system can solve problems that are getting closer to having real-life applications. All in all, they have come a long way since the first hype-filled announcement.

Over time, my skepticism has waxed and waned. Although I didn't really trust their demonstrations, D-Wave's papers, which usually made more limited claims, seemed pretty solid. Now, there is a new data point to add to the list, with a paper claiming to show that the D-Wave machine cannot be doing classical simulated annealing.

Once again in English, please

Annealing is a process where you carefully and slowly allow a physical system to relax. As it's relaxing, it will carefully arrange itself so that it has the lowest possible amount of energy, called the ground state.

An example of this is a set of magnets. Each magnet can arrange itself so that it is either pointing up or down. At high temperatures (in other words, way above the ground state), each magnet arranges itself randomly, either pointing up or down. And, because the energy difference between the two states is small, they flip up and down in response to tiny changes in the local magnetic field (caused by neighboring magnets flipping).

As the magnets are cooled down, the rate of flipping slows down; however, as that occurs, the influence of the direction of the neighboring magnets becomes more substantial. In this case, where the only magnetic field is due to the magnets themselves, they start to pair up to minimize their total joint energy and reduce the total magnetic field to zero. That is, each magnet tries to minimize its energy with respect to its surroundings.

This process, called annealing, can also be simulated using a computer, and is a pretty nice way to solve some very complicated problems. But simulated annealing is, in principle, no faster than any other classical mechanism for calculating solutions to problems—although it may be more convenient to implement and optimize, and be faster in practice.

The relevance to D-Wave?

D-Wave's computer pretty much does this. They use superconducting loops to generate tiny individual magnets where the orientation depends on the direction in which the current circulates. These tiny magnets are coupled to each other so that their orientations influence each other.

To solve a problem with this bunch of magnets, however, you need to rewrite the problem so that its solution is the magnets' lowest energy state. To get there, the magnet's orientations are initialized in a well understood way, and the system is placed in the ground state for that configuration. Slowly, the environment around the magnets and coupling among them is modified so that it resembles the problem. If that is done correctly, the magnets may change their orientations, but never leave the ground state. By reading out their final orientations, you obtain the solution to the problem you wanted.

If this happens to be a classical process—which we just discussed above—then this is no faster than a classical computer. However, if the magnetics are behaving in a quantum manner, then it might be a quantum computer and could be faster.

So, is it quantum or not?

According to a recent paper in Nature Communications, the D-Wave device is not doing classical simulated annealing. Which, unfortunately, means exactly that. It tells us what it isn't, but doesn't tell us what it is.

To go into this a little more deeply, the researchers analyzed how the coupling between the magnets created a ground state. The layout of the hardware consists of four inner magnets arranged in a diamond (so each magnet is coupled directly to two others). Each of these is coupled to one additional magnet, but those are not coupled to each other. This configuration appears to be set up such that the four inner magnets always have the same orientation, while the outer magnets are free to arrange themselves as they see fit.

This results in a rather strange set of 17 possible ground states, most of which can be reached in steps of single flips of magnets. Except for the last, which requires that all four inner magnets flip at the same time.

In a classical simulation, the set of magnets can sample many different states. But, if by chance it happens to flip into this last ground state, it becomes trapped there. Furthermore, once it is there, the outer magnets become trapped in a single state too, because all other configurations have higher energy. Of course, once in this isolated state, it can also get out by flipping all four inner magnets, but the isolation and lack of noise (the outer magnets can't flip either) mean that it is, in some sense, less likely to flip out of the state than into it.

In the quantum description of these events, this doesn't happen. After setting up the ground state, we start trying to move to the solution state (by varying the environment). As soon as we do that, the ground state splits up, and the isolated state where things get stuck raises up in energy, away from the ground state. Since everything is kept in the ground state, it is no surprise that we find that the probability of entering the isolated state reduces sharply.

But, notice that this is different from the classical case. In the classical case, there was no way to break up the ground state. In other words, the energetic descriptions of the classical and quantum ground states are not the same, and it is no surprise that they give two different results.

Why the difference?

At heart, this difference was inevitable. When you get right down to it, we live in a quantum world, and if you are careful enough, that will shine through. In some ways, this shows how sloppy our thinking about the whole thing is. When we think of simulated annealing, or anything else like this, we imagine a purely classical or a purely quantum system. In reality, things are a lot more messy, with some aspects remaining classical and others showing their quantum nature.

What these results show is that we can't treat the D-Wave optimizer as a purely classical device. But, whether that actually means anything in practice is very hard to tell. One of the things that I admire about this, though, is the way the research output is slowly piling up, with much of the evidence being positive for D-Wave. It is the best way to answer critics, both professional and amateur: keep generating evidence and data.

Promoted Comments

What sort of problems could be solved by using/reading the ground state of magnets?

Any kind of problem where you have to take into account many variables and minimize some overall cost function. For example, let's say we are designing a suspension bridge. There is a single support tower in the middle, the body of the bridge itself, and the major support truss that goes from the top of the tower to the ends of the bridge body. The major truss is attached to the body by a series of minor vertical trusses. The placement of the minor trusses is not trivial - we can vary their spacing, thickness, material, tension, and so forth. There can be hundreds of minor trusses. An analytical solution may be completely intractable. A simple gradient descent optimizer will probably be unstable, and likely won't even find the most optimal solution, instead getting stuck in a local minimum somewhere. Simulated annealing can avoid local minima, and can generally deal with huge variable spaces.

As long as you can abstractly represent the original problem as a distribution of magnetic spins, this is applicable to anything.

Chris Lee
Chris writes for Ars Technica's science section. A physicist by day and science writer by night, he specializes in quantum physics and optics. He Lives and works in Eindhoven, the Netherlands. Emailchris.lee@arstechnica.com

57 Reader Comments

What I don't understand is how the functioning of this device can be so poorly understood. Why does all the news about it make it sound like it's a mystery box, and not something that a bunch of engineers at D-Wave designed and built?

I realize I have asked a question that opens the door to lots of clever jokes about the uncertain nature of quantum mechanics.

The way it's described, you'd think D-Wave reverse-engineered the thing from a UFO crash site or something.

"As soon as we do that, the ground state splits up ... Since everything is kept in the ground state ... In the classical case, there was no way to break up the ground state."

I fell over these words. I understand what each one means, but not these sentences. I understood the earlier explanation of what a ground state is, i.e. a low energy situation, but I can't make the jump to conceiving of what the quote is describing.

Like a good little boy, I even googled 'split the ground state', but all I got was stuff about hyperfine splitting of hydrogen atoms, which wasn't much help.

There are two different approaches to the problem in the article, classical simulated annealing and quantum annealing. In simulated annealing you start with a system that has all the correct interactions between its parts, but is at high temperature. By cooling it down slowly you hope for the final state of the system to be the ground state (i.e. the correct solution). Note that because the fundamental description of interactions between its parts never change, the ground state (whether we fall into it or not) also never changes.

By contrast, in quantum annealing you start with the system in the known ground state of a known problem with an easy solution, the "problem" being encoded in the interactions of the systems various parts. Instead of lowering temperature (in quantum annealing one will always want the temperature to be low and stay there) you slowly change the *interaction* between the parts, which can change the ground state continuously, including splitting any "degenerate" (same energy) solutions that were all previously valid ground states and change them so that now some of them become the new ground state, but others do not. (This is analogous to the fine and hyperfine splitting you ran into, where two states that were previously of the same energy split into ones of different energy.) In quantum annealing one hopes to stay in the ground state (of a constantly changing problem) the whole time so that eventually you change the interaction of the parts to exactly describe the problem you are actually interested in solving, and thus have the right answer.

And that's the difference. In simulated annealing the ground state never changes, you just aren't in it until the end. In quantum annealing the ground state is changing constantly but you stay in it, you just aren't describing the problem you want until the end. (Assuming both systems actually find the correct solution: simulated annealing can easily land in not-quite-the-right solution, and in quantum annealing the interactions must change very slowly or ruin the "adiabatic" property that keeps the system in the ground state.) In practice, there are always thermal effects (i.e. from non-zero temperature) and there are reasons to suspect these might dominate the intended quantum effects, turning the D-Wave system into a very expensive classical simulated annealing machine -- thus the test to see if one can detect uniquely quantum behavior in an example problem.

(And if you're like me, you're wondering to what degree it can be called "simulated annealing" if the temperature of the D-Wave system is essentially constant, but it turns out that the way they change the interactions between parts changes the ratio between energy and temperature, which is actually the relevant quantity. We just usually call it temperature in simulated annealing because the energy scale never changes.)

What sort of problems could be solved by using/reading the ground state of magnets?

Finding prime factors of large numbers is one. The traveling salesman problem is another.

The traveling salesman problem is NP complete, and does not fall in the complexity class BQP that we believe describes the power of quantum computers. You can find "local" solutions to the traveling salesman problem, but you can find those with genetic or differential algorithms on a classical computer as well.

What sort of problems could be solved by using/reading the ground state of magnets?

Finding prime factors of large numbers is one. The traveling salesman problem is another.

The traveling salesman problem is NP complete, and does not fall in the complexity class BQP that we believe describes the power of quantum computers. You can find "local" solutions to the traveling salesman problem, but you can find those with genetic or differential algorithms on a classical computer as well.

Yeah, my mistake. Funnily enough, even though the travelling salesman problem is probably not speed up by quantum computing, it still gets touted as though it does (google supposedly purchased one for network routing solutions, which sounds very similar).

What I don't understand is how the functioning of this device can be so poorly understood. Why does all the news about it make it sound like it's a mystery box, and not something that a bunch of engineers at D-Wave designed and built?

I realize I have asked a question that opens the door to lots of clever jokes about the uncertain nature of quantum mechanics.

The "is it classical or quantum" argument is really a smoke screen covering the real argument: "is it any faster than just running classical computations on a GPU cluster?"

The answer so far has been "no".

Someone correct me if I'm wrong, but I believe this is how the debate over the quantum nature lays out:

It would be possible to run a processor similar to DWave's that has no quantum effects going on within it (just little magnets flipping around like described in the article). Since a final result is based on a statistical analysis of repeated runs of the same initial condition, you might get very similar answers using a classical setup. But the quantum setup should converge on a solution faster and with more accuracy than the classical setup would. But since we don't have a "perfectly classical" or "perfectly quantum" version to compare this to, then it's actually kind of hard to figure out how much quantumness is going on in there just by looking at the results.

So the guys referenced in the article have actually shown that hard thing -- that the results they are seeing in some special case could only be the result of some kind of quantum effect occurring (probably "only" is overstated... they are probably applying a statistical test like "95% certainty that the results are consistent with quantum and not classical effects").

But, as I opened this post with, there's still lots of people whose response to that is "cool... but are they getting faster computation out of it?"

It should be noted that few (if any) of the critics I've read online think that DWave isn't pushing the science of quantum computing forward with what they're doing. There's just a question of whether they have a product that is superior to competitors (i.e. GPU clusters) or whether this is just cool science. But everyone seems to agree it's at least cool science.

For those interested in the question of whether the D-Wave machine offers any speedup, Scott Aaronson has a very nice blog post about it HERE. The post is rather long, and discusses the question of whether anything quantum is going on as well. He also links to THIS post by Alex Selby, which seems to indicate that the D-Wave machine is outperformed by classical approaches.

Can't they solve a known problem, such as traveling salesman problem, for a large number of cities whose solution is already known, and see IF it can solve it and how long did it take it to solve it compared to a modern classical computer? And what is the maximum number of cities it can handle compared to a classical computer?. Thanks.

What sort of problems could be solved by using/reading the ground state of magnets?

Finding prime factors of large numbers is one. The traveling salesman problem is another.

The traveling salesman problem is NP complete, and does not fall in the complexity class BQP that we believe describes the power of quantum computers. You can find "local" solutions to the traveling salesman problem, but you can find those with genetic or differential algorithms on a classical computer as well.

Yeah, my mistake. Funnily enough, even though the travelling salesman problem is probably not speed up by quantum computing, it still gets touted as though it does (google supposedly purchased one for network routing solutions, which sounds very similar).

What makes the traveling salesman problem hard is the "return to the start" part. The problem of optimizing a path between two separate points in a graph/map/network (those three are all equivalent mathematically) has a much more reasonable O(n). Such path optimization problems might still be susceptible to quantum speedup, but it's fundamentally a different problem than travelling salesman.

What sort of problems could be solved by using/reading the ground state of magnets?

Finding prime factors of large numbers is one. The traveling salesman problem is another.

The traveling salesman problem is NP complete, and does not fall in the complexity class BQP that we believe describes the power of quantum computers. You can find "local" solutions to the traveling salesman problem, but you can find those with genetic or differential algorithms on a classical computer as well.

I thought that adiabatic quantum computing was (in theory) capable of solving QMA problems, and not merely BQP. QMA is thought to be a superset of NP in a similar manner as BQP is thought to be a superset of P. Am I wrong?

What sort of problems could be solved by using/reading the ground state of magnets?

Finding prime factors of large numbers is one. The traveling salesman problem is another.

The traveling salesman problem is NP complete, and does not fall in the complexity class BQP that we believe describes the power of quantum computers. You can find "local" solutions to the traveling salesman problem, but you can find those with genetic or differential algorithms on a classical computer as well.

I thought that adiabatic quantum computing was (in theory) capable of solving QMA problems, and not merely BQP. QMA is thought to be a superset of NP in a similar manner as BQP is thought to be a superset of P. Am I wrong?

I'm not certain about the details regarding traveling salesman, but there certainly are examples of problems outside BQP which quantum computers give a speedup over classical computers on. However, that speedup can still leave the problem intractable. For example, Grover's algorithm runs in O(sqrt(N)), whereas the best classical algorithm for attacking the same problem runs in O(N). N, however, is exponential in the number of input bits to the function, so both algorithms run in superpolynomial time and are therefore "intractable". The nice thing about problems in BQP is that they are in a sense tractable (or efficiently solvable) on a quantum computer.

What sort of problems could be solved by using/reading the ground state of magnets?

Finding prime factors of large numbers is one. The traveling salesman problem is another.

The traveling salesman problem is NP complete, and does not fall in the complexity class BQP that we believe describes the power of quantum computers. You can find "local" solutions to the traveling salesman problem, but you can find those with genetic or differential algorithms on a classical computer as well.

I thought that adiabatic quantum computing was (in theory) capable of solving QMA problems, and not merely BQP. QMA is thought to be a superset of NP in a similar manner as BQP is thought to be a superset of P. Am I wrong?

I'm not certain about the details regarding traveling salesman, but there certainly are examples of problems outside BQP which quantum computers give a speedup over classical computers on. However, that speedup can still leave the problem intractable. For example, Grover's algorithm runs in O(sqrt(N)), whereas the best classical algorithm for attacking the same problem runs in O(N). N, however, is exponential in the number of input bits to the function, so both algorithms run in superpolynomial time and are therefore "intractable". The nice thing about problems in BQP is that they are in a sense tractable (or efficiently solvable) on a quantum computer.

You lost me at the "N is exponential in the number of input bits" part. This is certainly the first I've heard that Grover's doesn't provide practical speedup in any situation. Is there a specific application that you're talking about?

What sort of problems could be solved by using/reading the ground state of magnets?

Finding prime factors of large numbers is one. The traveling salesman problem is another.

The traveling salesman problem is NP complete, and does not fall in the complexity class BQP that we believe describes the power of quantum computers. You can find "local" solutions to the traveling salesman problem, but you can find those with genetic or differential algorithms on a classical computer as well.

Yeah, my mistake. Funnily enough, even though the travelling salesman problem is probably not speed up by quantum computing, it still gets touted as though it does (google supposedly purchased one for network routing solutions, which sounds very similar).

I was just poking around reading about adiabatic quantum computing. Interestingly, it's been proposed that adiabatic computing could specifically speed up the calculation of PageRank (not just "processes similar to PageRank, but specifically citing PageRank). Presumably, this supposes you could design an algorithm that maps "most pertinent" to "lowest energy state".

What sort of problems could be solved by using/reading the ground state of magnets?

Finding prime factors of large numbers is one. The traveling salesman problem is another.

The traveling salesman problem is NP complete, and does not fall in the complexity class BQP that we believe describes the power of quantum computers. You can find "local" solutions to the traveling salesman problem, but you can find those with genetic or differential algorithms on a classical computer as well.

I thought that adiabatic quantum computing was (in theory) capable of solving QMA problems, and not merely BQP. QMA is thought to be a superset of NP in a similar manner as BQP is thought to be a superset of P. Am I wrong?

I'm not certain about the details regarding traveling salesman, but there certainly are examples of problems outside BQP which quantum computers give a speedup over classical computers on. However, that speedup can still leave the problem intractable. For example, Grover's algorithm runs in O(sqrt(N)), whereas the best classical algorithm for attacking the same problem runs in O(N). N, however, is exponential in the number of input bits to the function, so both algorithms run in superpolynomial time and are therefore "intractable". The nice thing about problems in BQP is that they are in a sense tractable (or efficiently solvable) on a quantum computer.

You lost me at the "N is exponential in the number of input bits" part. This is certainly the first I've heard that Grover's doesn't provide practical speedup in any situation. Is there a specific application that you're talking about?

Grover's algorithm is often explained as an algorithm for searching an unordered list, where N is the number of elements in the list. Perhaps a more accurate description is that it takes a boolean function (that maps n input bits to 1 output bit) and outputs an input bit string for which the function evaluates to 1. For the search interpretation, each possible input string is the address for an element in the list and the function evaluates to 1 for all addresses holding the element you're searching for. In this case, N=2^n, since the number of possible bit strings goes up exponentially with the number of bits. Either way you look at it, Grover's only gives a quadratic speedup, which is not enough to make an intractable problem tractable the way these things are usually defined.

Edit: On the point of practical speedup, the speedup provided by Grover's certainly strikes me as practical in that it can make existing problems run faster. I simply wished to point out that it does not offer the dramatic speedup usually expected of quantum algorithms.

Grover's algorithm is often explained as an algorithm for searching an unordered list, where N is the number of elements in the list. Perhaps a more accurate description is that it takes a boolean function (that maps n input bits to 1 output bit) and outputs an input bit string for which the function evaluates to 1. For the search interpretation, each possible input string is the address for an element in the list and the function evaluates to 1 for all addresses holding the element you're searching for. In this case, N=2^n, since the number of possible bit strings goes up exponentially with the number of bits. Either way you look at it, Grover's only gives a quadratic speedup, which is not enough to make an intractable problem tractable the way these things are usually defined.

Edit: On the point of practical speedup, the speedup provided by Grover's certainly strikes me as practical in that it can make existing problems run faster. I simply wished to point out that it does not offer the dramatic speedup usually expected of quantum algorithms.

I get it now. Another way of saying it is that is that for Grover's Algorithm there is a relatively narrow range of N where a quantum computer would be usefully faster than a classical computer. For smaller Ns, classical would be fast enough. For larger Ns, quantum computers still wouldn't be fast enough. While those two statements are each always true at some N for quantum computers, the point is that other problem/algorithm combinations can result in much wider ranges of N where the quantum computer is a win.

Yeah, my mistake. Funnily enough, even though the travelling salesman problem is probably not speed up by quantum computing, it still gets touted as though it does (google supposedly purchased one for network routing solutions, which sounds very similar).

Solving the traveling salesman problem would not be sped up by quantum computing, but nobody doing any kind of engineering work is trying to solve it. Heck, when I learned this stuff in school one of the major points being taught was how to explain to your boss why it would be a complete waste for you to try, and an even bigger waste of time for them to try to replace you with a different employee who could.

I'm certain that google's interest is in the potential for speeding up heuristic algorithms for Traveling Salesmen, which while not NP-complete can still be quite time consuming. Whether D-Wave's current implementation actually does vs. custom classical hardware is an open question, though I'm willing to cut them slack on that given how classical hardware's significant engineering head start. Customers, obviously, have no reason to feel the same way.

But an employee of D-Wave, Dr. Suzanne Gildert, who has a Ph.D.in quantum computing, clearly said in a talk at Birmingham University in England that this very problem was solvable on a quantum computer! Whether she meant their D-Wave computer or a Universal Gate Model QM, I don't know. What gives?. You could probably enquire from Dr. Gildert herself, through her e-mail. Good luck. Thanks.